The Tetractys

by Juan Icon

The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself. The sum of 1, 2, 3, and 4, is 10.

Hence they called both the decad and the tetrad as every number.

The decad was a number in energy, but the tetrad was a number in capacity.

The sum of these four numbers 1, 2, 3, 4 made up the tetractys, in which all harmonic ratios are included.

  • 4 to 1 is a quadruple ratio. It forms the symphony bisdiapason.
  • 3 to 2 is a sesquialter ratio. It forms the symphony diapente.
  • 4 to 329 3 is a sesquitertian ratio. If forms the symphony diatessaron.
  • 2 to 1 is a duple ratio. It forms the diapason.

The Pythagoreans greatly venerated the the tetractys.

Theo of Smyrna,[106] showed how many tetractys there are. He said:

"The tetractys was principally honored by the Pythagoreans because all symphonies are found to exist within it, and also because it appears to contain the nature of all things.”

Hence their oath was:

“Not by Pythagoras who invented the tetractys, which contains the fountain and root of everlasting nature.”

The tetractys, therefore, is seen in the composition of the first numbers 1. 2. 3. 4.

But the second tetractys arises from the increase by multiplication of even and odd numbers beginning from the monad.

First Tetractys

Of these, the monad is assumed as the first because it is the principle of all even, odd, and evenly-odd numbers. Its nature is simple.

But the three successive numbers receive their composition according to the even and the odd because every number is not alone even, nor alone odd.

Second Tetractys

Hence the even and the odd receive two tetractys, according to multiplication.

  • The even numbers receive in a duple ratio; for 2 is the first of even numbers, and increases from the monad by duplication.
  • But the odd number is increased in a triple ratio; for 3 is the first of odd numbers, and is itself increased from the monad by triplication.

Hence the monad is common to both these, being itself even and odd.

The second number, however, in even and double numbers is 2; but in odd and triple numbers 3. The third among even numbers is 4; but among odd numbers is 9. And the fourth among even numbers is 8; but among odd numbers is 27.

{ 1. 2. 4.  8. }
{ 1. 3. 9. 27. }

In these numbers the more perfect ratios of symphonies are found. In these also a tone is comprehended.

The monad, however, contains the productive principle of a point.

But the second numbers 2 and 3 contain the principle of a side, since they are incomposite, and first, are measured by the monad, and naturally measure a right line.

The third terms are 4 and 9, which are in power a square superficies, since they are equally equal. And the fourth terms 8 and 27 being equally equally equal, are in power a cube.

Hence from these 331 numbers, and this tetractys, the increase takes place from a point to a solid.

For a side follows after a point, a superficies after a side, and a solid after a superficies. In these numbers also, Plato in the Timæus constitutes the soul. But the last of these seven numbers, i. e. 27, is equal to all the numbers that precede it; for 1 + 2 + 3 + 4 + 8 + 9 = 27.

There are, therefore, two tetractys of numbers, one of which subsists by addition, but the other by multiplication, and they comprehend musical, geometrical, and arithmetical ratios, from which also the harmony of the universe consists.

Third Tetractys

But the third tetractys is that which according to the same analogy or proportion comprehends the nature of all magnitude.

For what the monad was in the former tetractys, that a point is in this.

What the numbers 2 and 3, which are in power a side, were in the former tetractys, that the extended species of a line, the circular and the right, are in this; the right line indeed subsisting in conformity to the even number, since it is terminated[107] by two points; but the circular in conformity to the odd number, because it is comprehended by one line which has no end. But what in the former tetractys the square numbers 4 and 9 were, that the two-fold 332 species of planes, the rectilinear and the circular, are in this.

And what the cube numbers 8 and 27 were in the former, the one being an even, but the other an odd number, that the two solids, one of which has a hollow superficies, as the sphere and the cylinder, but the other a plane superficies, as the cube and pyramid, are in this tetractys. Hence, this is the third tetractys, which gives completion to every magnitude, from a point, a line, a superficies, and a solid.

Fourth Tetractys: The Four Elements

The fourth tetractys is of the simple bodies fire, air, water, and earth, which have an analogy according to numbers.

For what the monad was in the first tetractys, that fire is in this.

But the duad is air, the triad is water, and the tetrad is earth.

For such is the nature of the elements according to tenuity and density of parts. Hence fire has to air the ratio of 1 to 2; but to water, the ratio of 1 to 3; and to earth, the ratio of 1 to 4.

In other respects also they are analogous to each other.

Fifth Tetractys: Simple Bodies

The fifth tetractys is of the figures of the simple bodies.

  • The pyramid is the shape of fire
  • The octaedron is the shape of air
  • The icosaedron is the shape of water
  • The cube is the shape of earth

Sixth Tetractys: Life

The sixth tetractys is of things rising into existence through the vegetative life.

The seed is analogous to the monad and a point.

But if it increases in length it is analogous to the duad and a line. If in breadth, to the triad and a superficies; but if in thickness, to the tetrad and a solid.

Seventh Tetractys: Society

The seventh tetractys is of communities. The principle as the monad, is man. The duad is a house. The triad a street; and the tetrad a city. A nation consists of these.

These are the material and sensible tetractys.

Eighth Tetractys: Sense and Intellect

These are in the powers which form a judgment of things material and sensible, and which are of a certain intelligible nature – intellect, science, opinion, and sense.

Intellect, indeed, corresponds in its essence to the monad.

Science corresponds to the duad because science is the science of a certain thing.

Opinion subsists between science and ignorance.

Sense is as the tetrad because the touch which is common to all the senses being fourfold, all the senses energize according to contact.

Ninth Tetractys: Soul and Body

The ninth tetractys is that from which the animal is composed, the soul and the body. For the parts of the soul, indeed, are the rational, the irascible, and the epithymetic, or that which desires external good; and the fourth is the body in which the soul subsists.

Tenth Tetractys: Seasons

The tenth tetractys is of the seasons of the year, 334 through which all things rise into existence, viz. the spring, the summer, the autumn, and the winter.

Eleventh Tetractys

The eleventh is of the ages of man:

  • the infant
  • the child
  • the adult
  • the old adult

Hence there are eleven tetractys.

  1. The first is that which subsists according to the composition of numbers.
  2. The second is the multiplication of numbers.
  3. The third subsists according to magnitude.
  4. The fourth is of the simple bodies.
  5. The fifth is of shapes.
  6. The sixth is of things rising into existence through the vegetative life.
  7. The seventh is of communities.
  8. The eighth is the judicial power.
  9. The ninth is of the parts of the animal.
  10. The tenth is of the seasons of the year.
  11. The eleventh is of the ages of man.

All of them however are proportional to each other.

A monad manifests as the following:

  1. A monad
  2. A monad
  3. A point
  4. A fire
  5. A pyramid
  6. A seed
  7. A man
  8. Intellect

Thus, if the first tetractys is 1, 2, 3, 4:

  • The second is the monad, a side, a square, and a cube.
  • The third is a point, a line, a superficies, and a solid.
  • The fourth is fire, air, water, earth.
  • The fifth the pyramid, the octaedron, the icosaedron, and the cube.
  • The sixth, seed, length, breadth and depth.
  • The seventh, man, a house, a street, a city.
  • The eighth, intellect, science, opinion, sense.
  • The ninth, the rational, the irascible, and the epithymetic parts, and the body.
  • The tenth, the spring, summer, autumn, winter.
  • The eleventh, the infant, the lad, the man, and the old man.

The universe is composed from these tetractys. It is perfect and being elegantly arranged in geometrical, harmonical, and arithmetical proportion. It comprehends every power, all the nature of number, every magnitude, and every simple and composite body.

But it is perfect, because all things are the parts of it, but it is not itself the part of any thing.

Hence, the Pythagoreans first used the before-mentioned oath, and asserted that “all things are assimilated to number.”

P. 111. This number is the first that partakes of every number, and when divided in every possible way, receives the power of the numbers subtracted, and of those that remain.

Because 6 consists of 1, 2 and 3, the two first of which are the principles of all number, and also because 2 and 3 are the first even and odd, which are the sources of all the species of numbers; the number 6 may be said to partake of every number.

In what Iamblichus afterwards adds, I suppose he alludes to 6 being a perfect number and therefore equal to all its parts.